$4$Introduction

Objective functions plays an important role to validate and compare the performance of optimiza$-$

tion algorithms. Many benchmark or test functions have been documented in various publications,

but there isn't a universally accepted list of benchmark functions. The most effective test functions

should possess a range of distinct characteristics, allowing for unbiased testing of new algorithms.

In $[1],$ a rich set of $175$ benchmark functions for unconstrained optimization problems with diverse

properties are presented. In the course, as you know, we made a tutorial session for

$f_{175}(x)=0\mathrm{.}25x_1^4-0\mathrm{.}5x_1^2+0\mathrm{.}1x_1+0\mathrm{.}5x_2^2$

subject to $-2x_{i}2,$ for $i=1,2,$ by using Newton$-$Raphson and Hestenes$-$Stiefel algorithms.

Main Task

Write a MATLAB program to find the minimum of the function $f_{cd}(*)$ where $cd=ab(mod50)$ from

$,$ Appendix B$]$ by using

Newton$-$Raphson,

Hestenes$-$Stiefel,

Polak$-$Ribi$$re and

Fletcher$-$Reeves algorithms.

You are also strongly encouraged to use another relevant algorithm from the literature, which will

be rewarded with an extra $10$ points. If the function is not differentiable, use an approximation

proposed by yourself or an relevant approximation commonly used in the literature and write it

explicitly in your report. Repeat the main steps of your algorithms until the desired accuracy is

achieved, i$.$e$.$

$||gradf(x_k)||lon$ and $|f(x_{k+1})-f(x_k)|lon.$

Take THREE initial guess as $x_0{N}_n(0,1)(n-$dimensional vector having elements from standard

normal distribution by using randn function of MATLAB$)$ or $x_0{U}_n(x_{0,\text{m}\text{i}\text{n}},x_{0,\text{m}\text{a}\text{x}})(n-$dimensional

vector having elements from uniform distribution from the closed interval $x_{0,\text{m}\text{i}\text{n}},x_{0,\text{m}\text{a}\text{x}}$ where

$x_{0,\text{m}\text{i}\text{n}}$ and $x_{0,\text{m}\text{a}\text{x}}$ are specified for each function in $[1]$ by using rand function of MATLAB$).$ For

instance, if your problem is defined on $-2x_{i}2,$ for $i=1,2,$ you may consider to choose

$x_0{U}_n(-2,2).$ Take also the absolute error bound as $lon={10}^{-4}$ for every algorithm.

Shekel $5$ Problem $(55)($Dixon and Szeg$,1978)$

$mi{n}_{x}f(x)=-{\displaystyle \underset{i=1}{\overset{5}{}}}\frac{1}{{\displaystyle \underset{j=1}{\overset{4}{}}}(x_j-a_{ij}{)}^2+c_i}$

subject to $0x_{j}10,$jin$\{1,2,3,4\}$ with constants $a_{ij}$ and $c_j$ given in

Table $15$ below. There are five local minima and the global minimizer is

located at $x^{**}=(4\mathrm{.}00,4\mathrm{.}00,4\mathrm{.}00,4\mathrm{.}00)$ with $f(x^{**})$~~$-10\mathrm{.}1499$